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20 days and 35 nights – it’s just TOO long

Walking from our gate to baggage claim Sunday afternoon, we encountered a family of four stopped at the bottom of the stairs, surrounded by their bags, in the midst of a huge family hug-complete with smiles and tears.  The kids (the boy was about 5 and the girl was about 7) were finished with the hugging business before Mom and Dad and were waiting patiently.  In a bit, Dad stepped back and opened his arms really wide, and with a huge smile said, “I am so glad to see you all. We are finally altogether after being apart for 5 weeks this summer.”

The boy replied, in a most serious voice, “I know, Dad.  I haven’t seen you in 20 days.  It was so long!”

20 days

The Dad’s most perfect response to his child was, “Yes it was such a long time and I missed you so much.”  The math was perfect, as well, in the fact that it was a perfect opportunity to talk about how long 20 days are and how long 5 weeks feels.  Maybe not right at that moment, as they had to get their things at baggage claim, but perhaps in the car or on Monday afternoon…

And then there was Monday…Move into the College Dorm Day

We live on the other side of the country from the University my daughter attends-so we store her things with a local company.  Great service.  You buy their boxes.  You pack up your stuff.  They come pick it up and store it for the summer.  Then, like magic, they bring it back to you in the fall on move-in day.  Way better than renting a storage unit, renting a car, finding boxes, carrying the boxes up and down the stairs (if, like freshman year, you live in a building without an elevator), taking it to the storage place 3 minutes before they close-and it’s REALLY, REALLY dark and scary, and then trying to make your plane.  Love the guys who take care of the boxes!!!  We just show up and the boxes appear.  The guys who make this happen are awesome.

The confirmation email said, “We’ll be there between 10 am and 4 pm!”  Wait a minute.  Up until this time, we always had a 2-hour window for the delivery time.  Okay, we’ll be optimistic.  They are starting on her side of the campus and we’ll be in the morning.  So, we unpacked the 3 suitcases we brought with us-despite the missing dresser. We neatly stacked everything that belonged in the dresser on the desk. Hung up the towels in the bathroom.  Made the bed-minus the comforter and the pillows-which were in the boxes that had yet to show up…and waited and waited. It was a really long time before the boxes appeared.

The boxes arrived about 4:45 pm.

Although it had only been about 7 hours, it felt like days – like 20 days – like 5 weeks!!

While we were NOT patiently waiting, we talked about how much we despise waiting — waiting in line, waiting for boxes to arrive, waiting to get to that important day.  And, we talked about how sometimes we want time to slow down so that summer is NOT over, vacation is not ending.

We have a system for measuring time-seconds, minutes, hours, days, weeks, months, years, decades, centuries. Time flies.  Time crawls.  Time stands still.  Time rushes by.

We have tools for measuring time: clocks, calendars, stop watches, timers. And they are everywhere, like this clock I found above a restaurant on the Upper East Side of Manhattan…

clock

Yet, with all of these tools, placed throughout the world right before us, how is it that 20 days can be the same as 5 weeks?  Well, it is all about the context.  So talk with your kids about how time feels and how time has this cool math component-we can measure it, organize it, quantify it.  Yet, we still feel it.  Yes, 20 days to a 5 year old feels like forever.  And 20 days in a lifetime of 102 years might feel differently, or it might feel the same.  But, mathematically, the proportion of the days to the years of the life are vastly different.  But both are equally important to the person. So perhaps we should think about, and discuss, the feeling of time and the math of time and how, sometimes, we just don’t know everything.

Regardless of whether we have all of the answers, I hope you enjoy your time. And enjoy your math conversations!

Oh-and the dresser-well, the dresser arrived the next day…

Sequence, Counting, and Cardinal aspects of number words

We are one of those crazy districts that starts school in August, and at the beginning of August, no less.  Because of that, I have been doing a lot of reading and thinking about all stuff school.  One of the books I am reading is entitled Primary Mathematics:  Teaching for Understanding, written by Patrick Barmby, Lynn Bilsborough, Tony Harries and Steve Higgins.  It is a fabulous resource, and as noted on the back cover, the authors’ intent is “to support and develop teachers’ understanding of the key primary mathematics topics. ”

One of the first topics our brand new kindergarteners tackle is counting.  They must count and count and count and then count some more, as they practice saying the number words in sequence.  In Chapter 2, Barmby, et al. discuss how children acquire the spoken representation of number and what that path might look like. “When mistakes occur in the use of sequence numbers, these are usually related to children’s developing skills rather than particular misconceptions” (Barmby, et al., pg. 17).

To move through the Early Learning Counting Trajectory (the developmental sequence described by Douglas Clements and Julie Sarama), from a “reciter”-says number words in order to a “counter and producer”-counts objects, counts out a specific number of objects, & uses cardinality, kids must count. They must count things, lots of different things, lots of times:  Unifix cubes, pebbles, toy cars, pencils, shiny markers, kids in their class, stickers, sea shells, and dots on five- and ten-frame cards.  The kiddos must count objects and they must talk about what they are counting.

They should build trains of Unifix cubes and talk about how many cubes in each train; chat about which train of cubes is longer, which train is the shortest, and which two might be of the same length. Kindergarteners must create sets of objects – buttons, shells, dice-shaped erasers, plastic pumpkins – and then, count and re-count how many objects in each of the sets.

So, have young kids count and chat, and explore and investigate, and converse and discuss-it’s good for them.


Mistakes, Errors Patterns, and How Many Chairs?

Last week at Twitter Math Camp, Michael Pershan-of www.mathmistakes.org fame- gave a talk entitled Using Mistakes to Inspire Teaching.  He began his session with the following statement. “It’s not a waste of time to theorize deeply about student mistakes. In fact, I think it will enrich your teaching.  So, give it a shot.”  He then asked us to ponder:  where do mistakes come from and where, when, and how does an error pattern develop?

Such interesting questions to explore.  We spend lots of time and energy assuring and reassuring kids that it is “okay to make mistakes”-yet, we all HATE it when we make a mistake.  We don’t take much comfort from others saying that it is okay-as long as we learn from it-and all of the other platitudes that are offered.  However, it does encourage  me to wonder if we should be emphasizing mistakes like we do-turning it into this BIG deal-and that maybe we don’t need to be handling mistakes the way we are.  Dr. Johnston says that when you make a mistake, it means nothing more than that.  Fix it.  Learn from it.  So, perhaps we need to take a more low-key, natural approach to mistakes and errors.  It is, after all, part of building understanding, part of figuring stuff out, part of learning.  To quote Dr. Johnston (again), “errors usually happen at the edge of what we can do, when we are stretching into new territory-when we are learning.”  Aren’t errors and mistakes a regular component of everyday life, so maybe we should just treat it as part of what happens-no big deal?

Later on in his talk, Michael displayed some students’ work on exponents and his talking points focused on how we might categorize mistakes. He suggested that we think about them from the perspective of a specific response, “as if this were my student, how would I respond?” or through a more general lens, “why do students tend to do this?”  All of this builds on the context of his opening statement that theorizing about student mistakes enriches and inspires teaching.

So, what if mistakes were just a part of the conversation, the brainstorming, the sharing of ideas, the classroom discourse in a mathematics class?  What if errors and mistakes are just what Dr. Johnston says they are? What if they just became part of the learning conversation and the work that we do?  They could, in fact, inspire our teaching.  Encouraging us to organize instruction to embed the mistakes within the sequence and structure of learning.  This in turn offers kids multiple opportunities to build understanding, and ensures entry into the mathematics.  They might enrich our teaching by encouraging learning processes to be built in classrooms so that students become accustomed to ideas-those that work and those that don’t work so well-being part of the way learning happens. The use of error patterns could be examined and used to support the understanding and knowledge we want kids to acquire.

In light of these ideas, consider the following problem of the day found on page 52 in the publication English Language Learners in the Mathematics Classroom by D. Coggins, D. Kravin, G. Coates, and M. Carroll.

HOW MANY CHAIRS?  Chairs need to be set up in the multipurpose room for an assembly.  In this room, 23 rows of chairs will fit.  In each row, 34 chairs will fit.  How many chairs will fit in the room?  Show how you know in at least two ways.

Groups of learners worked on the task and the following is one of the solutions presented during the class conversation.

photo

How might the errors and mistake in this solution be used to enrich our teaching? How might we use this to further students’ understanding of multiplication strategies?  What role might the errors and mistakes take in the conversation to support understanding?  Is this part of an error pattern in the making?

Certainly lots to ponder about mistakes and errors, learning and kids, and the importance of our language as we get ready to launch the 2013-2014 school year.